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3.
Objectives State and explain the Fundamental Theorems of Calculus Use the ﬁrst fundamental theorem of calculus to ﬁnd deriva ves of func ons deﬁned as integrals. Compute the average value of an integrable func on over a closed interval.

4.
Outline Recall: The Evalua on Theorem a/k/a 2nd FTC The First Fundamental Theorem of Calculus Area as a Func on Statement and proof of 1FTC Biographies Diﬀeren a on of func ons deﬁned by integrals “Contrived” examples Erf Other applica ons

5.
The deﬁnite integral as a limit Deﬁni on If f is a func on deﬁned on [a, b], the deﬁnite integral of f from a to b is the number ∫ b ∑n f(x) dx = lim f(ci ) ∆x a n→∞ i=1 b−a where ∆x = , and for each i, xi = a + i∆x, and ci is a point in n [xi−1 , xi ].

6.
The deﬁnite integral as a limit Theorem If f is con nuous on [a, b] or if f has only ﬁnitely many jump discon nui es, then f is integrable on [a, b]; that is, the deﬁnite ∫ b integral f(x) dx exists and is the same for any choice of ci . a

7.
Big time Theorem Theorem (The Second Fundamental Theorem of Calculus) Suppose f is integrable on [a, b] and f = F′ for another func on F, then ∫ b f(x) dx = F(b) − F(a). a

13.
Outline Recall: The Evalua on Theorem a/k/a 2nd FTC The First Fundamental Theorem of Calculus Area as a Func on Statement and proof of 1FTC Biographies Diﬀeren a on of func ons deﬁned by integrals “Contrived” examples Erf Other applica ons

22.
The area function in general Let f be a func on which is integrable (i.e., con nuous or with ﬁnitely many jump discon nui es) on [a, b]. Deﬁne ∫ x g(x) = f(t) dt. a The variable is x; t is a “dummy” variable that’s integrated over. Picture changing x and taking more of less of the region under the curve. Ques on: What does f tell you about g?

37.
Another Big Time Theorem Theorem (The First Fundamental Theorem of Calculus) Let f be an integrable func on on [a, b] and deﬁne ∫ x g(x) = f(t) dt. a If f is con nuous at x in (a, b), then g is diﬀeren able at x and g′ (x) = f(x).

106.
Meet the MathematicianJames Gregory Sco sh, 1638-1675 Astronomer and Geometer Conceived transcendental numbers and found evidence that π was transcendental Proved a geometric version of 1FTC as a lemma but didn’t take it further

107.
Meet the MathematicianIsaac Barrow English, 1630-1677 Professor of Greek, theology, and mathema cs at Cambridge Had a famous student

109.
Meet the MathematicianGottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathema cian invented calculus 1672–1676 published in papers 1684 and 1686

110.
Diﬀerentiation and Integration asreverse processes Pu ng together 1FTC and 2FTC, we get a beau ful rela onship between the two fundamental concepts in calculus. Theorem (The Fundamental Theorem(s) of Calculus) I. If f is a con nuous func on, then ∫ d x f(t) dt = f(x) dx a So the deriva ve of the integral is the original func on.

111.
Diﬀerentiation and Integration asreverse processes Pu ng together 1FTC and 2FTC, we get a beau ful rela onship between the two fundamental concepts in calculus. Theorem (The Fundamental Theorem(s) of Calculus) II. If f is a diﬀeren able func on, then ∫ b f′ (x) dx = f(b) − f(a). a So the integral of the deriva ve of is (an evalua on of) the original func on.

112.
Outline Recall: The Evalua on Theorem a/k/a 2nd FTC The First Fundamental Theorem of Calculus Area as a Func on Statement and proof of 1FTC Biographies Diﬀeren a on of func ons deﬁned by integrals “Contrived” examples Erf Other applica ons

128.
Why use 1FTC? Ques on Why would we use 1FTC to ﬁnd the deriva ve of an integral? It seems like confusion for its own sake.

129.
Why use 1FTC? Ques on Why would we use 1FTC to ﬁnd the deriva ve of an integral? It seems like confusion for its own sake. Answer Some func ons are diﬃcult or impossible to integrate in elementary terms.

130.
Why use 1FTC? Ques on Why would we use 1FTC to ﬁnd the deriva ve of an integral? It seems like confusion for its own sake. Answer Some func ons are diﬃcult or impossible to integrate in elementary terms. Some func ons are naturally deﬁned in terms of other integrals.

137.
Other functions deﬁned by integrals The future value of an asset: ∫ ∞ FV(t) = π(s)e−rs ds t where π(s) is the proﬁtability at me s and r is the discount rate. The consumer surplus of a good: ∫ q∗ ∗ CS(q ) = (f(q) − p∗ ) dq 0 where f(q) is the demand func on and p∗ and q∗ the equilibrium price and quan ty.

145.
Summary Func ons deﬁned as integrals can be diﬀeren ated using the ﬁrst FTC: ∫ d x f(t) dt = f(x) dx a The two FTCs link the two major processes in calculus: diﬀeren a on and integra on ∫ F′ (x) dx = F(x) + C